Correlation of CMB with large-scale structure. II. Weak lensing

We investigate the correlation of gravitational lensing of the cosmic microwave background (CMB) with several tracers of large-scale structure, including luminous red galaxies (LRGs), quasars, and radio sources. The lensing field is reconstructed based on the CMB maps from the Wilkinson Microwave Anisotropy Probe (WMAP) satellite; the LRGs and quasars are observed by the Sloan Digital Sky Survey (SDSS); and the radio sources are observed in the NRAO VLA Sky Survey (NVSS). Combining all three large-scale structure samples, we find evidence for a positive cross correlation at the $2.5\sigma$ level ($1.8\sigma$ for the SDSS samples and $2.1\sigma$ for NVSS); the cross correlation amplitude is $1.06\pm 0.42$ times that expected for the WMAP cosmological parameters. Our analysis extends other recent analyses in that we carefully determine bias-weighted redshift distribution of the sources, which is needed for a meaningful cosmological interpretation of the detected signal. We investigate contamination of the signal by galactic emission, extragalactic radio and infrared sources, thermal and kinetic Sunyaev-Zel’dovich effects, and the Rees-Sciama effect, and find all of them to be negligible.

Figure 1

The response factor $R_l$ of Eq. (15). This is the calibration relating the (mean of the) reconstructed map $\mathbf{v}$ to the underlying convergence factor $\kappa$. Note that it depends on scale.

Figure 2

The CMB lensing map $\nabla ·{\mathbf{v}}^{(TT)}$ in Galactic Molleweide projection, degraded to $40\mathrm{arc}\min$ resolution. The gray region is rejected by the Kp052 mask. The scale runs from $-2\times {10}^8$ (black) to $+2\times {10}^8$ (white). The map is dominated by noise (both instrumental and due to the finite number of primary CMB modes available for the reconstruction).

Figure 3

The LSS-convergence cross-spectra. Each column represents a different large-scale structure tracer (LRGs, quasars, and NVSS), and each row represents a different frequency combination (the first row, TT, is frequency-averaged; the other rows are VV, VW, and WW). The horizontal dashed line is zero, and the thick line shows the theoretical signal for the fiducial cosmology and the best-fit LRG bias/redshift distribution (see Paper I). The last data point in each plot is actually two $l$-bins that have been combined so that their error bars do not dominate the vertical scale of the plot.

Figure 4

The pseudo-$C_l$ power spectrum of the reconstructed lensing map (Eq. (27)) for the Kp052 cut. These power spectra are dominated by noise at all scales. The top panel shows the power spectrum of the full reconstruction, whereas the other panels show the power spectra of the ${\mathbf{v}}^{VV}$, ${\mathbf{v}}^{VW}$, and ${\mathbf{v}}^{WW}$ lensing maps. The solid line shows the data, and the dotted lines show five simulations. Note that with the Kp052 sky cut the simulations reproduce the power spectrum of the data to within several percent. Similar plots for the Kp05 cut show significant excess power.

Figure 5

The LSS-convergence cross-spectrum $C_l^{g\kappa }$ obtained by rotating $\mathbf{v}$ 90 degrees before computing the cross-correlation. This should be zero in the absence of systematics. The three panels show the LRGs, quasars, and the NVSS sample. The solid lines show the signal expected for the unrotated maps in the WMAP cosmology. The last data point in each plot is actually two $l$-bins that have been combined so that their error bars do not dominate the vertical scale of the plot.

Figure 6

Constraints on the frequency dependence of the lensing signal. The horizontal axis is the ratio of thermodynamic temperatures in the V and W bands, which is 1 for the CMB. The vertical axis is the ${\chi }^2$ obtained by fitting the values of the lensing amplitude $A$ in the VV, VW, and WW frequency combinations.